Philosophical Significance

Category theory challenges philosophers in two ways, which are not necessarily mutually exclusive. On the one hand, it is certainly the task of philosophy to clarify the general epistemological and ontological status of categories and categorical methods, both in the practice of mathematics and in the foundational landscape. On the other hand, philosophers and philosophical logicians can employ category theory and categorical logic to explore philosophical and logical problems. I now discuss these challenges, briefly, in turn.

Category theory is now a common tool in the mathematician's toolbox; that much is clear. It is also clear that category theory organizes and unifies much of mathematics. (See for instance Mac Lane 1971, 1998 or Pedicchio & Tholen 2004.) No one will deny these simple facts.

Doing mathematics in a categorical framework is almost always radically different from doing it in a set-theoretical framework (the exception being working with the internal language of a Boolean topos; whenever the topos is not Boolean, then the main difference lies in the fact that the logic is intuitionistic). Hence, as is often the case when a different conceptual framework is adopted, many basic issues regarding the nature of the objects studied, the nature of the knowledge involved, and the nature of the methods used have to be reevaluated. We will take up these three aspects in turn.

Two facets of the nature of mathematical objects within a categorical framework have to be emphasized. First, objects are always given in a category. An object exists in and depends upon an ambient category. Furthermore, an object is characterized by the morphisms going in it and/or the morphisms coming out of it. Second, objects are always characterized up to isomorphism (in the best cases, up to a unique isomorphism). There is no such thing, for instance, as the natural numbers. However, it can be argued that there is such a thing as the concept of natural numbers. Indeed, the concept of natural numbers can be given unambiguously, via the Dedekind-Peano-Lawvere axioms, but what this concept refers to in specific cases depends on the context in which it is interpreted, e.g., the category of sets or a topos of sheaves over a topological space. It is hard to resist the temptation to think that category theory embodies a form of structuralism, that it describes mathematical objects as structures since the latter, presumably, are always characterized up to isomorphism. Thus, the key here has to do with the kind of criterion of identity at work within a categorical framework and how it resembles any criterion given for objects which are thought of as forms in general. One of the standard objections presented against this view is that if objects are thought of as structures and only as abstract structures, meaning here that they are separated from any specific or concrete representation, then it is impossible to locate them within the mathematical universe. (See Hellman 2003 for a standard formulation of the objection, McLarty 1993, Awodey 2004, Landry & Marquis 2005, Shapiro 2005, Landry 2011, Linnebo & Pettigrew 2011, McLarty 2011 for relevant material on the issue.)

A slightly different way to make sense of the situation is to think of mathematical objects as types for which there are tokens given in different contexts. This is strikingly different from the situation one finds in set theory, in which mathematical objects are defined uniquely and their reference is given directly. Although one can make room for types within set theory via equivalence classes or isomorphism types in general, the basiccriterion of identity within that framework is given by the axiom of extensionality and thus, ultimately, reference is made to specific sets. Furthermore, it can be argued that the relation between a type and its token is not represented adequately by the membership relation. A token does not belong to a type, it is not an element of a type, but rather it is an instance of it. In a categorical framework, one always refers to a token of a type, and what the theory characterizes directly is the type, not the tokens. In this framework, one does not have to locate a type, but tokens of it are, at least in mathematics, epistemologically required. This is simply the reflection of the interaction between the abstract and the concrete in the epistemological sense (and not the ontological sense of these latter expressions.) (See Ellerman 1988, Marquis 2000, Marquis 2006, Marquis 2013.)

The history of category theory offers a rich source of information to explore and take into account for an historically sensitive epistemology of mathematics. It is hard to imagine, for instance, how algebraic geometry and algebraic topology could have become what they are now without categorical tools. (See, for instance, Carter 2008, Corfield 2003, Krömer 2007, Marquis 2009, McLarty 1994, McLarty 2006.) Category theory has lead to reconceptualizations of various areas of mathematics based on purely abstract foundations. Moreover, when developed in a categorical framework, traditional boundaries between disciplines are shattered and reconfigured; to mention but one important example, topos theory provides a direct bridge between algebraic geometry and logic, to the point where certain results in algebraic geometry are directly translated into logic and vice versa. Certain concepts that were geometrical in origin are more clearly seen as logical (for example, the notion of coherent topos). Algebraic topology also lurks in the background. On a different but important front, it can be argued that the distinction between mathematics and metamathematics cannot be articulated in the way it has been. All these issues have to be reconsidered and reevaluated.

Moving closer to mathematical practice, category theory allowed for the development of methods that have changed and continue to change the face of mathematics. It could be argued that category theory represents the culmination of one of deepest and most powerful tendencies in twentieth century mathematical thought: the search for the most general and abstract ingredients in a given situation. Category theory is, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures. When used to characterize a specific mathematical domain, category theory reveals the frame upon which that area is built, the overall structure presiding to its stability, strength and coherence. The structure of this specific area, in a sense, might not need to rest on anything, that is, on some solid soil, for it might very well be just one part of a larger network that is without any Archimedean point, as if floating in space. To use a well-known metaphor: from a categorical point of view, Neurath's ship has become a spaceship.

Still, it remains to be seen whether category theory should be “on the same plane,” so to speak, as set theory, whether it should be taken as a serious alternative to set theory as a foundation for mathematics, or whether it is foundational in a different sense altogether. (That this very question applies even more forcefully to topos theory will not detain us.)

Lawvere from early on promoted the idea that a category of categories could be used as a foundational framework. (See Lawvere 1964, 1966.) This proposal now rests in part on the development of higher-dimensional categories, also called weak n-categories. (See, for instance Makkai 1998.) The advent of topos theory in the seventies brought new possibilities. Mac Lane has suggested that certain toposes be considered as a genuine foundation for mathematics. (See Mac Lane 1986.) Lambek proposed the so-called free topos as the best possible framework, in the sense that mathematicians with different philosophical outlooks might nonetheless agree to adopt it. (See Couture & Lambek 1991, 1992, Lambek 1994.) He has recently argued that there is no topos that can thoroughly satisfy a classical mathematician. (See Lambek 2004.) (For more on the various foundational views among category theorists, see Landry & Marquis 2005.)

Arguments have been advanced for and against category theory as a foundational framework. (Blass 1984 surveys the relationships between category theory and set theory. Feferman 1977, Bell 1981, and Hellman 2003 argue against category theory. See Marquis 1995 for a quick overview and proposal and McLarty 2004 and Awodey 2004 for replies to Hellman 2003.) This matter is further complicated by the fact that the foundations of category theory itself have yet to be clarified. For there may be many different ways to think of a universe of higher-dimensional categories as a foundations for mathematics. An adequate language for such a universe still has to be presented together with definite axioms for mathematics. (See Makkai 1998 for a short description of such a language. A different approach based on homotopy theory but with closed connections with higher-dimensional categories has been proposed by Voevodsky et al. and is being vigorously pursued. See the book Homotopy Type Theory, by Awodey et al. 2013.)

It is an established fact that category theory is employed to study logic and philosophy. Indeed, categorical logic, the study of logic by categorical means, has been under way for about 30 years now and still vigorous. Some of the philosophically relevant results obtained in categorical logic are:

• The hierarchy of categorical doctrines: regular categories, coherent categories, Heyting categories and Boolean categories; all these correspond to well-defined logical systems, together with deductive systems and completeness theorems; they suggest that logical notions, including quantifiers, arise naturally in a specific order and are not haphazardly organized;
• Joyal's generalization of Kripke-Beth semantics for intuitionistic logic to sheaf semantics (Lambek & Scott 1986, Mac Lane & Moerdijk 1992);
• Coherent and geometric logic, so-called, whose practical and conceptual significance has yet to be explored (Makkai & Reyes 1977, Mac Lane & Moerdiejk 1992, Johnstone 2002, Caramello 2011b, 2012a);
• The notions of generic model and classifying topos of a theory (Makkai & Reyes 1977, Boileau & Joyal 1981, Bell 1988, Mac Lane & Moerdijk 1992, Johnstone 2002, Caramello 2012b);
• The notion of strong conceptual completeness and the associated theorems (Makkai & Reyes 1977, Butz & Moerdijk 1999, Makkai 1981, Pitts 1989, Johnstone 2002);
• Geometric proofs of the independence of the continuum hypothesis and other strong axioms of set theory (Tierney 1972, Bunge 1974, Freyd 1980, 1987, Blass & Scedrov 1983, 1989, 1992, Mac Lane & Moerdijk 1992);
• Models and development of constructive mathematics (see bibliography below);
• Synthetic differential geometry, an alternative to standard and non-standard analysis (Kock 1981, Bell 1998, 2001, 2006);
• The construction of the so-called effective topos, in which every function on the natural numbers is recursive (McLarty 1992, Hyland 1982, 1991, Van Oosten 2002, Van Oosten 2008);
• Categorical models of linear logic, modal logic, fuzzy sets, and general higher-order type theories (Reyes 1991, Reyes & Zawadoski 1993, Reyes & Zolfaghari 1991, 1996, Makkai & Reyes 1995, Ghilardi & Zawadowski 2002, Rodabaugh & Klement 2003, Jacobs 1999, Taylor 1999, Johnstone 2002, Blute & Scott 2004, Awodey & Warren 2009, Awodey et. al. 2013);
• A graphical syntax called “sketches” (Barr & Wells 1985, 1999, Makkai 1997a, 1997b, 1997c, Johnstone 2002).
• Quantum logic, the foundations of quantum physics and quantum field theory (Abramsky & Duncan 2006, Heunen et. al. 2009, Baez & Stay 2010, Baez & Lauda 2011, Coecke 2011, Isham 2011, Döring 2011).

Categorical tools in logic offer considerable flexibility, as is illustrated by the fact that almost all the surprising results of constructive and intuitionistic mathematics can be modeled in a proper categorical setting. At the same time, the standard set-theoretic notions, e.g. Tarski's semantics, have found natural generalizations in categories. Thus, categorical logic has roots in logic as it was developed in the twentieth century, while at the same time providing a powerful and novel framework with numerous links to other parts of mathematics.

Category theory also bears on more general philosophical questions. From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics. Similar remarks can be made when we turn to ontology, in particular formal ontology: the part/whole relation, boundaries of systems, ideas of space, etc. Ellerman (1988) has bravely attempted to show that category theory constitutes a theory of universals, one having properties radically different from set theory, which is also seen as a theory of universals. Moving from ontology to cognitive science, MacNamara & Reyes (1994) have tried to employ categorical logic to provide a different logic of reference. In particular, they have attempted to clarify the relationships between count nouns and mass terms. Other researchers are using category theory to study complex systems, cognitive neural networks, and analogies. (See, for instance, Ehresmann & Vanbremeersch 1987, 2007, Healy 2000, Healy & Caudell 2006, Arzi-Gonczarowski 1999, Brown & Porter 2006.) Finally, philosophers of science have turned to category theory to shed a new light on issues related to structuralism in science. (See, for instance, Brading & Landry 2006, Bain 2013, Lam & Wüthrich forthcoming.)

Category theory offers thus many philosophical challenges, challenges which will hopefully be taken up in years to come.

• –––, 2004, “An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism”, Philosophia Mathematica, 12: 54–64.
• –––, 2006, Category Theory, Oxford: Clarendon Press.
• –––, 2007, “Relating First-Order Set Theories and Elementary Toposes”, The Bulletin of Symbolic, 13 (3): 340–358.
• –––, 2008, “A Brief Introduction to Algebraic Set Theory”, The Bulletin of Symbolic, 14 (3): 281–298.
• Awodey, S., et al., 2013, Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program.
• Awodey, S. & Butz, C., 2000, “Topological Completeness for Higher Order Logic”, Journal of Symbolic Logic, 65 (3): 1168–1182.
• Awodey, S. & Reck, E. R., 2002, “Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic”, History and Philosophy of Logic, 23 (1): 1–30.
• –––, 2002, “Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics”, History and Philosophy of Logic, 23 (2): 77–94.
• Awodey, S. & Warren, M., 2009, “Homotopy theoretic Models of Identity Types”, Mathematical Proceedings of the Cambridge Philosophical Society, 146 (1): 45–55.
• Baez, J., 1997, “An Introduction to n-Categories”, Category Theory and Computer Science, Lecture Notes in Computer Science (Volume 1290), Berlin: Springer-Verlag, 1–33.
• Baez, J. & Dolan, J., 1998a, “Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes”, Advances in Mathematics, 135: 145–206.
• –––, 1998b, “Categorification”, Higher Category Theory (Contemporary Mathematics, Volume 230), Ezra Getzler and Mikhail Kapranov (eds.), Providence: AMS, 1–36.
• –––, 2001, “From Finite Sets to Feynman Diagrams”, Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.
• Baez, J. & Lauda, A.D., 2011, “A Pre-history of n-Categorical Physics”, Deep Beauty: Understanding the Quantum World Through Mathematical Innovation, H. Halvorson, ed., Cambridge: Cambridge University Press, 13–128.
• Baez, J. & May, P. J., 2010, Towards Higher Category Theory, Berlin: Springer.
• Baez, J. & Stay, M., 2010, “Physics, Topology, Logic and Computation: a Rosetta Stone”, New Structures for Physics (Lecture Notes in Physics 813), B. Coecke (ed.), New York, Springer: 95–172.
• Baianu, I. C., 1987, “Computer Models and Automata Theory in Biology and Medecine”, in Witten, Matthew, Eds. Mathematical Modelling, Vol. 7, 1986, chapter 11, Pergamon Press, Ltd., 1513–1577.
• Bain, J., 2013, “Category-theoretic Structure and Radical Ontic Structural Realism”, Synthese, 190: 1621–1635.
• Barr, M. & Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.
• –––, 1999, Category Theory for Computing Science, Montreal: CRM.
• Batanin, M., 1998, “Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories”, Advances in Mathematics, 136: 39–103.
• Bell, J. L., 1981, “Category Theory and the Foundations of Mathematics”, British Journal for the Philosophy of Science, 32: 349–358.
• –––, 1982, “Categories, Toposes and Sets”, Synthese, 51 (3): 293–337.
• –––, 1986, “From Absolute to Local Mathematics”, Synthese, 69 (3): 409–426.
• –––, 1988, “Infinitesimals”, Synthese, 75 (3): 285–315.
• –––, 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press.
• –––, 1995, “Infinitesimals and the Continuum”, Mathematical Intelligencer, 17 (2): 55–57.
• –––, 1998, A Primer of Infinitesimal Analysis, Cambridge: Cambridge University Press.
• –––, 2001, “The Continuum in Smooth Infinitesimal Analysis”, Reuniting the Antipodes — Constructive and Nonstandard Views on the Continuum (Synthese Library, Volume 306), Dordrecht: Kluwer, 19–24.
• –––, 2005, “The Development of Categorical Logic”, in Handbook of Philosophical Logic(Volume 12), 2nd ed., D.M. Gabbay, F. Guenthner (eds.), Dordrecht: Springer, pp. 279–362.
• Birkoff, G. & Mac Lane, S., 1999, Algebra, 3^rd ed., Providence: AMS.
• Blass, A., 1984, “The Interaction Between Category Theory and Set Theory”, in Mathematical Applications of Category Theory (Volume 30), Providence: AMS, 5–29.
• Blass, A. & Scedrov, A., 1983, “Classifying Topoi and Finite Forcing”, Journal of Pure and Applied Algebra, 28: 111–140.
• –––, 1989, Freyd's Model for the Independence of the Axiom of Choice, Providence: AMS.
• –––, 1992, “Complete Topoi Representing Models of Set Theory”, Annals of Pure and Applied Logic , 57 (1): 1–26.
• Blute, R. & Scott, P., 2004, “Category Theory for Linear Logicians”, in Linear Logic in Computer Science, T. Ehrhard, P. Ruet, J-Y. Girard, P. Scott, eds., Cambridge: Cambridge University Press, 1–52.
• Boileau, A. & Joyal, A., 1981, “La logique des topos”, Journal of Symbolic Logic, 46 (1): 6–16.
• Borceux, F., 1994, Handbook of Categorical Algebra, 3 volumes, Cambridge: Cambridge University Press.
• Brading, K. & Landry, E., 2006, “Scientific Structuralism: Presentation and Representation”, Philosophy of Science, 73: 571–581.
• Brown, R. & Porter, T., 2006, “Category Theory: an abstract setting for analogy and comparison”, What is Category Theory?, G. Sica, ed., Monza: Polimetrica: 257–274.
• Bunge, M., 1974, “Topos Theory and Souslin's Hypothesis”, Journal of Pure and Applied Algebra, 4: 159–187.
• –––, 1984, “Toposes in Logic and Logic in Toposes”, Topoi, 3 (1): 13–22.
• Caramello, O., 2011, “A Characterization Theorem for Geometric Logic”, Annals of Pure and Applied Logic,162, 4: 318–321.
• –––, 2012a, “Universal Models and Definability”, Mathematical Proceedings of the Cambridge Philosophical Society, 152 (2): 279–302.
• –––, 2012b, “Syntactic Characterizations of Properties of Classifying Toposes”, Theory and Applications of Categories, 26 (6): 176–193.
• Carter, J., 2008, “Categories for the working mathematician: making the impossible possible”, Synthese, 162 (1): 1–13.
• Cheng, E. & Lauda, A., 2004, Higher-Dimensional Categories: an illustrated guide book, available at: http://cheng.staff.shef.ac.uk/guidebook/index.html
• Cockett, J. R. B. & Seely, R. A. G., 2001, “Finite Sum-product Logic”, Theory and Applications of Categories (electronic), 8: 63–99.
• Coecke, B., 2011, “A Universe of Processes and Some of its Guises”, Deep Beauty: Understanding the Quantum World through Mathematical Innovation, Cambridge: Cambridge University Press: 129–186.
• Couture, J. & Lambek, J., 1991, “Philosophical Reflections on the Foundations of Mathematics”, Erkenntnis, 34 (2): 187–209.
• –––, 1992, “Erratum:”Philosophical Reflections on the Foundations of Mathematics“”, Erkenntnis, 36 (1): 134.
• Crole, R. L., 1994, Categories for Types, Cambridge: Cambridge University Press.
• Dieudonné, J. & Grothendieck, A., 1960 [1971], Éléments de Géométrie Algébrique, Berlin: Springer-Verlag.
• Döring, A., 2011, “The Physical Interpretation of Daseinisation”, Deep Beauty: Understanding the Quantum World through Mathematical Innovation, Cambridge: Cambridge University Press: 207-238.
• Ehresmann, A. & Vanbremeersch, J.-P., 2007, Memory Evolutive Systems: Hierarchy, Emergence, Cognition, Amsterdam: Elsevier
• –––, 1987, “Hierarchical Evolutive Systems: a Mathematical Model for Complex Systems”, Bulletin of Mathematical Biology, 49 (1): 13–50.
• Eilenberg, S. & Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.
• Eilenberg, S. & Mac Lane, S., 1942, “Group Extensions and Homology”, Annals of Mathematics, 43: 757–831.
• –––, 1945, “General Theory of Natural Equivalences”, Transactions of the American Mathematical Society, 58: 231–294.